Note: when the symbol ∘\circ is used, it denotes traditional right-to-left order of composition. For those who prefer the left-to-right order, the symbol ;; may be used in place of ∘\circ. Further discussion of this should go to the nForum page here.

where jj is the Yoneda embedding. Profunctors F:Cop×C→Set\mathbf{F} : C^{op} \times C \to Set whose hom-adjunct is of the form C→FC→j[Cop,Set]C \stackrel{F}{\to} C \stackrel{j}{\to} [C^{op}, Set] for FF an ordinary functor are those in the inclusion of these ordinary functors into profunctors. So the hom-functor is the image of the identity functor under this inclusion.